No. Only you know this.
If you've forgotten it, you can reset it... but I think you need to start the whole process from scratch. Depends what bank your with, I guess.
I can't give you a definitive answer (though I'm sure someone can) but I remember a story from a few years ago when a chap changed his password to "barklaysiscrap" or somesuch and it was disallowed.
In a decent secure system (like banks) then "no they don't" - the PW string you enter is encrypted by a non-reversible algorithm and compared with the output generated in the same way when you first set up the PW.
I'd be less sure about some other sites - many of them seem to store your PW details.
For this reason I always use a different password for each important (eg banking) site and have one or two general ones that I use for less secure / less important stuff.
If you sign up for a web site that does NOT encrypt passwords then if you register for that web site with your "normal" email address and password then in effect you are giving away the way for anyone to logon to your e
mail system.
The person who owns the web site could then log on to your email system and send out spam to all your contacts.
This is how many people's email systems are hacked in to.
As a minimum have a DIFFERTENT password for your email system that you dont use anywhere else.